# Prove that if $f:[a,+\infty [\longrightarrow \mathbb R$ is uniformly continuous, then $\lim_{x\to +\infty }f(x)=+\infty$ [closed]

I have to show that if $f:[a,+\infty [\longrightarrow \mathbb R$ is uniformly continuous, then $\lim_{x\to +\infty }f(x)=+\infty$. I spend very much time on it, and I can't conclude. How can I find a $\delta>0$ s.t. $|f(x)-f(y)|<\varepsilon$ if $|x-y|<\delta$ for all $\varepsilon>0$ ?

## closed as unclear what you're asking by GEdgar, Silvia Ghinassi, miracle173, Daniel FischerFeb 5 '16 at 18:24

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• A constant function $f:x \mapsto c$ is uniformly continuous but $\lim\limits_{x\to +\infty }f(x)=c$ is finite. – Watson Feb 5 '16 at 16:26
• it seems the question is wrong. – runaround Feb 5 '16 at 16:29
• Therefore, should be closed as "unclear what you are asking". – GEdgar Feb 5 '16 at 16:29

A constant function $f:x \mapsto c$ is uniformly continuous but $\lim\limits_{x\to +\infty }f(x)=c$ is finite.